Subgroups — #15673129

Order 2944.

420 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625097	C₂	1	1	yes	maximal, characteristic
2	C₂×#10625097	C₂	1	1	yes	maximal, characteristic
3	#13092183	C₂	1	4	yes	maximal
4	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#117	—	2	2	no	—
7	C₂₃×#117	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃	—	2	1	no	—
11	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
12	C₂×C₂²⋊Dic₂₃	C₂²	1	2	yes	—
13	#10625097	—	2	1	no	—
14	#10625097	—	2	1	no	—
15	#10625097	—	2	2	no	—
16	#10625097	—	2	1	no	—
17	#10625097	—	2	1	no	—
18	#10625097	C₄	1	4	yes	complemented
19	C₂×C₄⋊Dic₂₃	—	2	1	no	—
20	C₂×C₄⋊Dic₂₃	C₄	1	2	yes	complemented
21	C₂×C₄×Dic₂₃	—	2	1	no	—
22	C₂×C₄×Dic₂₃	C₄	1	2	yes	complemented
23	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	—	2	2	no	—
25	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₃	D₄	1	2	yes	—
34	C₂²×C₄×C₂₃	Q₈	1	2	yes	—
35	C₂²×Dic₂₃	—	2	2	no	—
36	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
37	C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₂²⋊C₄×C₂₃	—	2	2	no	—
39	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
40	C₄²×C₂₃	—	2	4	no	—
41	C₂²⋊Dic₂₃	—	2	2	no	—
42	C₂²⋊Dic₂₃	—	2	2	no	—
43	C₂²⋊Dic₂₃	—	2	2	no	—
44	C₂²⋊Dic₂₃	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₂₃	—	2	4	no	—
46	C₄⋊Dic₂₃	—	2	2	no	—
47	C₄⋊Dic₂₃	—	2	2	no	—
48	C₄×Dic₂₃	—	2	4	no	—
49	C₄×Dic₂₃	—	2	2	no	—
50	C₄×Dic₂₃	—	2	2	no	—
51	C₂³×C₂₃	—	4	2	no	—
52	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
53	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
57	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂³×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₂₃	—	2	4	no	—
60	C₂×C₄×C₂₃	—	4	4	no	—
61	C₂×C₄×C₂₃	C₄²	1	2	yes	—
62	C₂×C₄×C₂₃	C₄²	1	2	yes	—
63	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
70	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
72	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
77	C₂×Dic₂₃	—	2	4	no	—
78	C₂×Dic₂₃	—	2	4	no	—
79	C₂×Dic₂₃	—	4	4	no	—
80	C₂×Dic₂₃	—	4	4	no	—
81	#1740	—	23	1	no	maximal, Sylow
82	C₂²×C₂₃	—	2	2	no	—
83	C₂²×C₂₃	—	2	2	no	—
84	C₂²×C₂₃	—	2	2	no	—
85	C₂²×C₂₃	—	4	1	no	—
86	C₂²×C₂₃	—	4	1	no	—
87	C₂²×C₂₃	—	4	1	no	—
88	C₂²×C₂₃	—	4	1	no	—
89	C₂²×C₂₃	C₂×C₄²	1	1	yes	characteristic
90	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₄×C₂₃	—	4	4	no	—
97	C₄×C₂₃	#95	1	2	yes	—
98	C₄×C₂₃	#95	1	2	yes	—
99	C₄×C₂₃	#95	1	2	yes	—
100	C₄×C₂₃	#95	1	2	yes	—
101	Dic₂₃	—	4	4	no	—
102	Dic₂₃	—	4	4	no	—
103	C₂×#117	—	23	1	no	—
104	C₂×#117	—	23	1	no	—
105	C₂×#117	D₂₃	1	1	yes	characteristic
106	#342	—	23	4	no	—
107	C₂×C₂₃	—	2	1	no	—
108	C₂×C₂₃	—	2	1	no	—
109	C₂×C₂₃	—	2	1	no	—
110	C₂×C₂₃	—	2	1	no	—
111	C₂×C₂₃	—	2	1	no	—
112	C₂×C₂₃	—	2	1	no	—
113	C₂×C₂₃	C₂×#95	1	1	yes	characteristic
114	C₂×C₂₃	#410	1	2	yes	—
115	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
116	#117	—	2	2	no	—
117	#117	—	23	4	no	—
118	#117	—	46	1	no	—
119	#117	—	46	1	no	—
120	#117	—	46	1	no	—
121	#117	—	46	2	no	—
122	#117	—	46	1	no	—
123	#117	Dic₂₃	1	4	yes	complemented
124	C₂×C₄⋊C₄	—	2	1	no	—
125	C₂×C₄⋊C₄	—	23	2	no	—
126	C₂×C₄⋊C₄	—	46	1	no	—
127	C₂×C₂²⋊C₄	—	23	2	no	—
128	C₂×C₂²⋊C₄	—	23	2	no	—
129	C₂×C₂²⋊C₄	C₂×D₂₃	1	2	yes	—
130	C₂×C₄²	—	2	1	no	—
131	C₂×C₄²	—	23	2	no	—
132	C₂×C₄²	—	46	1	no	—
133	C₂₃	#1740	1	1	yes	characteristic, complemented, Sylow
134	C₂⁴	C₂²×D₂₃	1	1	yes	characteristic
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	23	4	no	—
137	C₂²×C₄	—	46	2	no	—
138	C₂²×C₄	Dic₄₆	1	2	yes	—
139	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
140	C₂²×C₄	C₄×D₂₃	1	2	yes	—
141	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
142	C₂²×C₄	D₉₂	1	1	yes	characteristic
143	C₂²×C₄	D₉₂	1	1	yes	characteristic
144	C₂²×C₄	C₂×Dic₂₃	1	2	yes	—
145	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
147	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	46	2	no	—
150	C₄⋊C₄	—	46	2	no	—
151	C₄⋊C₄	—	46	4	no	—
152	C₂²⋊C₄	—	2	2	no	—
153	C₂²⋊C₄	—	23	4	no	—
154	C₂²⋊C₄	—	46	2	no	—
155	C₂²⋊C₄	—	46	2	no	—
156	C₂²⋊C₄	—	46	2	no	—
157	C₂²⋊C₄	C₂×Dic₂₃	1	4	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	46	2	no	—
160	C₄²	—	46	4	no	—
161	C₄²	—	46	2	no	—
162	C₂³	—	4	2	no	—
163	C₂³	C₂×Dic₄₆	1	1	yes	characteristic
164	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
165	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
166	C₂³	C₂×D₉₂	1	1	yes	characteristic
167	C₂³	C₂²×Dic₂₃	1	1	yes	characteristic
168	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
169	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	46	4	no	—
173	C₂×C₄	—	46	4	no	—
174	C₂×C₄	—	92	4	no	—
175	C₂×C₄	—	92	4	no	—
176	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
177	C₂×C₄	C₄×Dic₂₃	1	2	yes	—
178	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
179	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
180	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
181	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
182	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
183	C₂×C₄	C₄⋊Dic₂₃	1	2	yes	—
184	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
186	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
187	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
188	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
189	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
190	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
191	C₂×C₄	C₂²⋊Dic₂₃	1	2	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	1	no	—
196	C₂²	—	4	1	no	—
197	C₂²	—	4	1	no	—
198	C₂²	—	4	1	no	—
199	C₂²	C₂×C₄×Dic₂₃	1	1	yes	characteristic
200	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
201	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
202	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
203	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
204	C₂²	C₂×(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
205	C₂²	C₂×C₂²⋊Dic₂₃	1	1	yes	characteristic
206	C₄	—	4	4	no	—
207	C₄	—	92	4	no	—
208	C₄	—	92	4	no	—
209	C₄	C₂₃⋊#95	1	2	yes	—
210	C₄	C₂₃⋊#95	1	2	yes	—
211	C₄	C₂₃⋊#95	1	2	yes	—
212	C₄	C₂₃⋊#95	1	2	yes	—
213	C₂	—	2	1	no	—
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	2	1	no	—
218	C₂	—	2	1	no	—
219	C₂	C₂×C₂₃⋊#95	1	1	yes	characteristic
220	C₂	#13092390	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	(C₂×D₂₃)⋊C₄
commutator	C₂²×C₂₃	C₂×C₄²
Frattini	C₂⁴	C₂²×D₂₃
Fitting	C₂×C₂₃×#117	C₂
socle	C₂²×C₂₃	C₂×C₂²⋊C₄
prime radical, cosocle	C₂⁴×C₂₃	C₂³
hypercenter	C₂×#117	D₂₃
hypocenter	C₂₃	#1740
LCS	G ▷ C₂²×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#117

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. C₂₃×#117	C₄	semidirect
18. #10625097	C₄	semidirect
20. C₂×C₄⋊Dic₂₃	C₄	semidirect
22. C₂×C₄×Dic₂₃	C₄	semidirect
123. #117	Dic₂₃	semidirect
133. C₂₃	#1740	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#10625097	2
#13092183	2
C₂×C₂₃×#117	2
#1740	23